No. 52 March-A pril'1990 $3.95 T H E M TEe H CAL J 0 URN A L COPIA Object Oriented Programming First it was BASIC, then it was structures, now it's objects. C++ afi<;ionados feel, of course, that objects are so powerful, so encompassing that anything could be so defined. I hope they're not placing bets, because if they are, money's no object. C++ 2.0 page 8 An objective view of the newest C++. Training A Neural Network Now that you have a neural network what do you do with it? Part two of a fascinating series. Debugging C page 21 Pointers Using MEM Keep C fro111 (C)rashing your system. An AT Keyboard Interface Use an AT keyboard with your latest project. And More ... Understanding Logic Families EPROM Programming Speeding Up Your AT Keyboard ((CHAOS MADE TO ORDER~ Explore the Magnificent and Infinite World of Fractals with FRAC LS™ AN ELECTRONIC KALEIDOSCOPE OF NATURES GEOMETRYTM With FracTools, you can modify and play with any of the included images, or easily create new ones by marking a region in an existing image or entering the coordinates directly. Filter out areas of the display, change colors in any area, and animate the fractal to create gorgeous and mesmerizing images. Special effects include Strobe, Kaleidoscope, Stained Glass, Horizontal, Vertical and Diagonal Panning, and Mouse Movies. The most spectacular application is the creation of self-running Slide Shows. Include any PCX file from any of the popular "paint" programs. FracTools also includes a Slide Show Programming Language, to bring a higher degree of control to your shows.
{ Final version for JMA - Nov 2, 2017 } Möbius Bridges Carlo H. Séquin CS Division, University of California, Berkeley E-mail: [email protected] Abstract Key concepts and geometrical constraints are discussed that allow the construction of a usable bridge that is topological equivalent to a Möbius band. A multi-year search in publications and on the internet for real-world bridges that meet these requirements has not identified a single clean construction that warrants the designation “Möbius bridge,” but a few promising designs can be found. Several simple but practical designs are presented here. 1. Introduction July 2017 marks the twentieth installment of the annual Bridges conference [1], which elucidates the connections between mathematics and art, music, architecture, and many other cultural venues. This year the conference has been held in Waterloo, Canada. In its twenty-year history it has visited many places around the globe, including Seoul in South Korea, Coimbra in Portugal, Pécz in Hungary, and Leeuwarden in the Netherlands, the hometown of M.C. Escher. This conference series got started by Reza Sarhangi [2] at Southwestern College in Winfield, Kansas. After a few occurrences at this initial location, Reza Sarhangi and other core members of this conference started discussing the possibility of establishing some kind of a commemorative entity of the conference on the Winfield campus. Since Escher, Möbius, and Klein are among the heroes of this Math-Art community, suggestions included an Escher Garden, a Möbius Bridge, or a Klein Bottle House. This prompted me to study the feasibility of such entities; and over the following year, I developed some practical designs for bridges and buildings that follow the geometry of a Möbius band [3].
Ununfoldable Polyhedra with Convex Faces Marshall Bern¤ Erik D. Demainey David Eppsteinz Eric Kuox Andrea Mantler{ Jack Snoeyink{ k Abstract Unfolding a convex polyhedron into a simple planar polygon is a well-studied prob- lem. In this paper, we study the limits of unfoldability by studying nonconvex poly- hedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that \open" polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry [5, 12, 20, 24] is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge cutting of the polyhedron, and the resulting simple polygon is called an edge unfolding or net. While the ¯rst explicit description of this problem is by Shephard in 1975 [24], it has been implicit since at least the time of Albrecht Durer,Ä circa 1500 [11]. It is widely conjectured that every convex polyhedron has an edge unfolding. Some recent support for this conjecture is that every triangulated convex polyhedron has a vertex unfolding, in which the cuts are along edges but the unfolding only needs to be connected at vertices [10]. On the other hand, experimental results suggest that a random edge cutting of ¤Xerox Palo Alto Research Center, 3333 Coyote Hill Rd., Palo Alto, CA 94304, USA, email: [email protected].
Accepted manuscript for The Mathematical Intelligencer, ISSN: 0343-6993 (print version) ISSN: 1866-7414 (electronic version) The final publication is available at Springer via http://link.springer.com/article/10.1007/s00283-016-9630-9 DOI 10.1007/s00283-016-9630-9 Bridges: A World Community for Mathematical Art Kristóf Fenyvesi Mathematical Art Reborn: Academic Gathering or Festival of the Arts? This is not the first time the Mathematical Communities column has featured the Bridges Organization: the 2005 conference1, in the breathtaking Canadian Rocky Mountains at Banff, was described in these pages by Doris Schattschneider [Schattschneider, 2006], a regular Bridges participant and Escher-specialist. The 2005 conference saw the debut of Delicious Rivers, Ellen Maddow’s play on the life of Robert Ammann, a postal worker who discovered a number of aperiodic tilings.2 Marjorie Senechal, The Mathematical Intelligencer’s current editor-in-chief, served as Maddow's consultant.3 A theatre premiér at a conference on mathematics? A production performed by mathematicians, moonlighting as actors? But this is Bridges. A quick look at the 2005 conference relays the “essence” of this scientific and artistic “happening” resembling a first-rate festival of the arts. True to its title, Renaissance Banff, the 2005 Bridges gave all members of its community, whether based in the sciences or the arts, the feeling that they had helped bring about a genuine rebirth. I use “community” in its most complete sense—including adults, children, artists, university professors, art lovers and local people—for the wealth of conference activities could only be accomplished through the participation of each and every individual present.
Demaine, Demaine, Lubiw Courtesy of Erik D. Demaine, Martin L. Demaine, and Anna Lubiw. Used with permission. 1999 1 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Albers at Bauhaus, 1927–1928] 2 Circular Variation from Bauhaus [Albers at Bauhaus, 1927–1928] 3 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 4 Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. 5 “Black Hexagon” Demaine, Demaine, Fizel 2006 Courtesy of Erik Demaine, Martin Demaine, and Jenna Fizel. Used with permission. 6 Hyparhedra: Platonic Solids [Demaine, Demaine, Lubiw 1999] 7 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 8 “Computational Origami” Erik & Martin Demaine MoMA, 2008– Elephant hide paper ~9”x15”x7” Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Computational/. 9 Peel Gallery, Houston Nov. 2009 Demaine & Demaine 2009 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Limit/. 10 “Natural Cycles” Erik & Martin Demaine JMM Exhibition of Mathematical Art, San Francisco, 2010 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/NaturalCycles/. 11 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/BlindGlass/. Demaine & Demaine 2010 12 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 13 θ = 30° n = 16 Courtesy of Erik D.
Maria Wittendorff Din guide til MARRAKECH muusmann FORLAG Maria Wittendorff Din guide til MARRAKECH muusmann FORLAG INDHOLD 57 Musée Tiskiwin 97 Dar Bellarj 7 Forord 138 Restauranter & cafeer – Museum Bert Flint 98 Dar Moulay Ali 59 Heritage Museum 99 Comptoir des Mines 140 Det marokkanske køkken 62 Maison de la Photographie 100 Festivaler 8 En lang historie kort 64 Musée de Mouassine 102 David Bloch Gallery 143 Gueliz 65 Musée Boucharouite 102 Galerie 127 143 Grand Café de la Poste 12 Historiske 67 Musée de la Femme 103 Musée Mathaf Farid Belkahia 144 La Trattoria seværdigheder 68 Musée des Parfums 104 Maison Denise Masson 145 +61 70 Aman – Musée Mohammed VI 105 La Qoubba Galerie d’Art 146 Gaïa 14 El Koutoubia 71 Observatoire Astronomie 106 Street art 146 Amandine 16 Almoravide-kuplen – Atlas Golf 147 Le Loft 17 Bymur & byporte 147 Le 68 Bar à Vin 19 Jamaa el-Fna 148 Barometre 108 Riads & hoteller 22 Gnawa 149 L’Annexe 72 Haver & parker 24 De saadiske grave 110 Riad Z 150 Le Petit Cornichon 26 Arkitektur 74 Jardin Majorelle 111 Zwin Zwin Boutique Hotel & Spa 150 L’Ibzar 32 El Badi 77 Jardin Secret 112 Riad Palais des Princesses 151 Amal 35 Medersa Ben Youssef 79 Den islamiske have 113 Riad El Walaa 152 Café Les Négociants 37 El Bahia 80 Jardin Menara 113 Dar Annika 153 Al Fassia 39 Dar El Bacha 82 Jardin Agdal 114 Riad Houma 153 Patron de la Mer – Musée des Confluences 83 Anima Garden 114 Palais Riad Lamrani 154 Moncho’s House Café 41 Garverierne 84 Cyber Park 115 Riad Spa Azzouz 154 Le Warner 42 Mellah 85 Jardin des Arts 116 La Maison
STEM+VISUAL ART A Curricular Resource for K-12 Idaho Teachers a r t Drew Williams, M.A., Art Education Boise State University + Table of Contents Introduction 1 Philosophy 2 Suggestions 2 Lesson Plan Design 3 Tips for Teaching Art 4 Artist Catalogue 5 Suggestions for Classroom Use 9 Lesson Plans: K-3 10 Lesson Plans: 4-6 20 Lesson Plans: 6-9 31 Lesson Plans: 9-12 42 Sample Images 52 Resources 54 References 55 STEM+VISUAL ART A Curricular Resource for K-12 Idaho Teachers + Introduction: Finding a Place for Art in Education Art has always been an integral part of students’ educational experiences. How many can remember their first experiences as a child manipulating crayons, markers and paintbrushes to express themselves without fear of judgement or criticism? Yet, art is more than a fond childhood memory. Art is creativity, an outlet of ideas, and a powerful tool to express the deepest thoughts and dreams of an individual. Art knows no language or boundary. Art is always innovative, as each image bears the unique identity of the artist who created it. Unfortunately as many art educators know all too well, in schools art is the typically among the first subjects on the chopping block during budget shortfalls or the last to be mentioned in a conversation about which subjects students should be learning. Art is marginalized, pushed to the side and counted as an “if-we-have-time” subject. You may draw…if we have time after our math lesson. We will have time art in our class…after we have prepared for the ISAT tests.
Folding Concave Polygons Into Convex Polyhedra: the L-Shape
Rose- Hulman Undergraduate Mathematics Journal Folding concave polygons into convex polyhedra: The L-Shape Emily Dinana Alice Nadeaub Isaac Odegardc Kevin Hartshornd Volume 16, No. 1, Spring 2015 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics aUniversity of Washington Terre Haute, IN 47803 bUniversity of Minnesota c Email: [email protected] University of North Dakota d http://www.rose-hulman.edu/mathjournal Moravian College Rose-Hulman Undergraduate Mathematics Journal Volume 16, No. 1, Spring 2015 Folding concave polygons into convex polyhedra: The L-Shape Emily Dinan Alice Nadeau Issac Odegard Kevin Hartshorn Abstract. Mathematicians have long been asking the question: Can a given convex polyhedron can be unfolded into a polygon and then refolded into any other convex polyhedron? One facet of this question investigates the space of polyhedra that can be realized from folding a given polygon. While convex polygons are relatively well understood, there are still many open questions regarding the foldings of non-convex polygons. We analyze these folded realizations and their volumes derived from the polygonal family of ‘L-shapes,’ parallelograms with another parallelogram removed from a corner. We investigate questions of maximal volume, diagonalflipping, and topological connectedness and discuss the family of polyhedra that share a L-shape polygonal net. Acknowledgements: We gratefully acknowledge support from NSF grant DMS-1063070 and the 2012 Lafayette College Research Experience for Undergraduates, where the majority of this research was undertaken. We would like to thank our research advisor, Dr. Kevin Hartshorn, who helped us with his great ideas, feedback, problem solving abilities and support throughout the project.
SCIENCE • ART • TECHNOLOGY 24 JULY THROUGH 10 AUGUST 2008 < Rinus Roelofs, PROGRAMME OF EVENTS new design for artwork consisting of a single 24 JULY – 10 AUGUST, 2008 continuous surface! (Location: Boer) BRIDGES, an annual conference founded in 1998 and serving since then 4 metres long, 2.5 metres high, as an international platform for artists, scientists and scholars working on corten steel the interface of science, art and technology will be held this year in Leeuwarden, The Netherlands from 24 - 29 July 2008. The approximately 200 visitors to the conference from 25 different countries will be coming to Leeuwarden to participate in a broad program of mathematical connections in art, music, architecture and science. Associated with the conference is a highly varied programme Gerard Caris > Polyhedral Net of cultural activities open to the general public that will run through Structure # 1, 10 August 2008. During the Open Day held on 29 July 2008, you can 160 x 95 x 95 cm become acquainted with a number of scientists and artists whilst young and old can participate in various workshops on Grote Kerkstraat in Exhibitions in churches Leeuwarden. The winners of the Gateways to Fryslân Competition will be announced during the conference. The artists exhibiting their work have come from abstract mathematical backgrounds to create many different visual products. Some artists, such as Gerard Caris (1925), limit their oeuvre to a single theme. In his case, this is the pentagon. In the village of Zweins, this artist will be showing a selection of his work including his ‘Polyhedral Net Structure # 1’.
Islamic Domes of Crossed-Arches: Origin, Geometry and Structural Behavior
Islamic domes of crossed-arches: Origin, geometry and structural behavior P. Fuentes and S. Huerta Polytechnic University of Madrid, Spain ABSTRACT: Crossed-arch domes are one of the earliest types of ribbed vaults. In them the ribs are intertwined forming polygons. The earliest known vaults of this type are found in the Great Mosque of Córdoba in Spain built in the mid 10th century, though the type appeared later in places as far as Armenia or Persia. This has generated a debate on their possible origin; a historical outline is given and the different hypotheses are discussed. Geometry is a fundamental part and the different patterns are examined. Though geometry has been thoroughly studied in Hispanic-Muslim decoration, the geometry of domes has very rarely been considered. The geometrical patterns in plan will be examined and afterwards, the geometric problems of passing from the plan to the three-dimensional space will be considered. Finally, a discussion about the possible structural behaviour of these domes is sketched. Crossed-arch domes are a singular type of ribbed vaults. Their characteristic feature is that the ribs that form the vault are intertwined, forming polygons or stars, leaving an empty space in the centre. The fact that the earliest known vaults of this type are found in the Great Mosque of Córdoba, built in the mid 10th century, has generated a debate on their possible origin. The thesis that appears to have most support is that of the eastern origin. This article describes the different hypothesis, to then proceed with a discussion of the geometry.